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Theorem pellexlem3 35109
Description: Lemma for pellex 35113. To each good rational approximation of  ( sqr `  D
), there exists a near-solution. (Contributed by Stefan O'Rear, 14-Sep-2014.)
Assertion
Ref Expression
pellexlem3  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  ~<_  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } )
Distinct variable group:    x, D, y, z

Proof of Theorem pellexlem3
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnex 10502 . . . 4  |-  NN  e.  _V
21, 1xpex 6542 . . 3  |-  ( NN 
X.  NN )  e. 
_V
3 opabssxp 5017 . . 3  |-  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  C_  ( NN  X.  NN )
42, 3ssexi 4538 . 2  |-  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  e.  _V
5 simprl 756 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  a  e.  QQ )
6 simprrl 766 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  0  <  a )
7 qgt0numnn 14385 . . . . . . . 8  |-  ( ( a  e.  QQ  /\  0  <  a )  -> 
(numer `  a )  e.  NN )
85, 6, 7syl2anc 659 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (numer `  a )  e.  NN )
9 qdencl 14375 . . . . . . . 8  |-  ( a  e.  QQ  ->  (denom `  a )  e.  NN )
105, 9syl 17 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (denom `  a )  e.  NN )
118, 10jca 530 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
(numer `  a )  e.  NN  /\  (denom `  a )  e.  NN ) )
12 simpll 752 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  D  e.  NN )
13 simplr 754 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  -.  ( sqr `  D )  e.  QQ )
14 pellexlem1 35107 . . . . . . 7  |-  ( ( ( D  e.  NN  /\  (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  -.  ( sqr `  D )  e.  QQ )  ->  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0 )
1512, 8, 10, 13, 14syl31anc 1233 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0 )
16 simprrr 767 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) )
17 qeqnumdivden 14380 . . . . . . . . . . . 12  |-  ( a  e.  QQ  ->  a  =  ( (numer `  a )  /  (denom `  a ) ) )
1817oveq1d 6249 . . . . . . . . . . 11  |-  ( a  e.  QQ  ->  (
a  -  ( sqr `  D ) )  =  ( ( (numer `  a )  /  (denom `  a ) )  -  ( sqr `  D ) ) )
1918fveq2d 5809 . . . . . . . . . 10  |-  ( a  e.  QQ  ->  ( abs `  ( a  -  ( sqr `  D ) ) )  =  ( abs `  ( ( (numer `  a )  /  (denom `  a )
)  -  ( sqr `  D ) ) ) )
2019breq1d 4404 . . . . . . . . 9  |-  ( a  e.  QQ  ->  (
( abs `  (
a  -  ( sqr `  D ) ) )  <  ( (denom `  a ) ^ -u 2
)  <->  ( abs `  (
( (numer `  a
)  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) )
215, 20syl 17 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
( abs `  (
a  -  ( sqr `  D ) ) )  <  ( (denom `  a ) ^ -u 2
)  <->  ( abs `  (
( (numer `  a
)  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) )
2216, 21mpbid 210 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  ( abs `  ( ( (numer `  a )  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) )
23 pellexlem2 35108 . . . . . . 7  |-  ( ( ( D  e.  NN  /\  (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  ( abs `  (
( (numer `  a
)  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) )  ->  ( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) )
2412, 8, 10, 22, 23syl31anc 1233 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) )
2511, 15, 24jca32 533 . . . . 5  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
( (numer `  a
)  e.  NN  /\  (denom `  a )  e.  NN )  /\  (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) )
2625ex 432 . . . 4  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( ( a  e.  QQ  /\  (
0  <  a  /\  ( abs `  ( a  -  ( sqr `  D
) ) )  < 
( (denom `  a
) ^ -u 2
) ) )  -> 
( ( (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) ) )
27 breq2 4398 . . . . . 6  |-  ( x  =  a  ->  (
0  <  x  <->  0  <  a ) )
28 oveq1 6241 . . . . . . . 8  |-  ( x  =  a  ->  (
x  -  ( sqr `  D ) )  =  ( a  -  ( sqr `  D ) ) )
2928fveq2d 5809 . . . . . . 7  |-  ( x  =  a  ->  ( abs `  ( x  -  ( sqr `  D ) ) )  =  ( abs `  ( a  -  ( sqr `  D
) ) ) )
30 fveq2 5805 . . . . . . . 8  |-  ( x  =  a  ->  (denom `  x )  =  (denom `  a ) )
3130oveq1d 6249 . . . . . . 7  |-  ( x  =  a  ->  (
(denom `  x ) ^ -u 2 )  =  ( (denom `  a
) ^ -u 2
) )
3229, 31breq12d 4407 . . . . . 6  |-  ( x  =  a  ->  (
( abs `  (
x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
)  <->  ( abs `  (
a  -  ( sqr `  D ) ) )  <  ( (denom `  a ) ^ -u 2
) ) )
3327, 32anbi12d 709 . . . . 5  |-  ( x  =  a  ->  (
( 0  <  x  /\  ( abs `  (
x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) )  <->  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )
3433elrab 3206 . . . 4  |-  ( a  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  <->  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )
35 fvex 5815 . . . . 5  |-  (numer `  a )  e.  _V
36 fvex 5815 . . . . 5  |-  (denom `  a )  e.  _V
37 eleq1 2474 . . . . . . 7  |-  ( y  =  (numer `  a
)  ->  ( y  e.  NN  <->  (numer `  a )  e.  NN ) )
3837anbi1d 703 . . . . . 6  |-  ( y  =  (numer `  a
)  ->  ( (
y  e.  NN  /\  z  e.  NN )  <->  ( (numer `  a )  e.  NN  /\  z  e.  NN ) ) )
39 oveq1 6241 . . . . . . . . 9  |-  ( y  =  (numer `  a
)  ->  ( y ^ 2 )  =  ( (numer `  a
) ^ 2 ) )
4039oveq1d 6249 . . . . . . . 8  |-  ( y  =  (numer `  a
)  ->  ( (
y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )
4140neeq1d 2680 . . . . . . 7  |-  ( y  =  (numer `  a
)  ->  ( (
( y ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  <->  ( (
(numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) )  =/=  0
) )
4240fveq2d 5809 . . . . . . . 8  |-  ( y  =  (numer `  a
)  ->  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  =  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) ) ) )
4342breq1d 4404 . . . . . . 7  |-  ( y  =  (numer `  a
)  ->  ( ( abs `  ( ( y ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) ) )  < 
( 1  +  ( 2  x.  ( sqr `  D ) ) )  <-> 
( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) )
4441, 43anbi12d 709 . . . . . 6  |-  ( y  =  (numer `  a
)  ->  ( (
( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  (
( y ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) )  <-> 
( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) )
4538, 44anbi12d 709 . . . . 5  |-  ( y  =  (numer `  a
)  ->  ( (
( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) )  <->  ( ( (numer `  a )  e.  NN  /\  z  e.  NN )  /\  ( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) )  =/=  0  /\  ( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) ) )
46 eleq1 2474 . . . . . . 7  |-  ( z  =  (denom `  a
)  ->  ( z  e.  NN  <->  (denom `  a )  e.  NN ) )
4746anbi2d 702 . . . . . 6  |-  ( z  =  (denom `  a
)  ->  ( (
(numer `  a )  e.  NN  /\  z  e.  NN )  <->  ( (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN ) ) )
48 oveq1 6241 . . . . . . . . . 10  |-  ( z  =  (denom `  a
)  ->  ( z ^ 2 )  =  ( (denom `  a
) ^ 2 ) )
4948oveq2d 6250 . . . . . . . . 9  |-  ( z  =  (denom `  a
)  ->  ( D  x.  ( z ^ 2 ) )  =  ( D  x.  ( (denom `  a ) ^ 2 ) ) )
5049oveq2d 6250 . . . . . . . 8  |-  ( z  =  (denom `  a
)  ->  ( (
(numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) )  =  ( ( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )
5150neeq1d 2680 . . . . . . 7  |-  ( z  =  (denom `  a
)  ->  ( (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  <->  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0 ) )
5250fveq2d 5809 . . . . . . . 8  |-  ( z  =  (denom `  a
)  ->  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  =  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) ) )
5352breq1d 4404 . . . . . . 7  |-  ( z  =  (denom `  a
)  ->  ( ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) )  <->  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) )
5451, 53anbi12d 709 . . . . . 6  |-  ( z  =  (denom `  a
)  ->  ( (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) )  <-> 
( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) )
5547, 54anbi12d 709 . . . . 5  |-  ( z  =  (denom `  a
)  ->  ( (
( (numer `  a
)  e.  NN  /\  z  e.  NN )  /\  ( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) ) )  <->  ( ( (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) ) )
5635, 36, 45, 55opelopab 4711 . . . 4  |-  ( <.
(numer `  a ) ,  (denom `  a ) >.  e.  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  <->  ( (
(numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  ( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  (
(denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) )
5726, 34, 563imtr4g 270 . . 3  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  ->  <.
(numer `  a ) ,  (denom `  a ) >.  e.  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } ) )
58 ssrab2 3523 . . . . . 6  |-  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  C_  QQ
59 simprl 756 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
a  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } )
6058, 59sseldi 3439 . . . . 5  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
a  e.  QQ )
61 simprr 758 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } )
6258, 61sseldi 3439 . . . . 5  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
b  e.  QQ )
6335, 36opth 4664 . . . . . . 7  |-  ( <.
(numer `  a ) ,  (denom `  a ) >.  =  <. (numer `  b
) ,  (denom `  b ) >.  <->  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )
64 simprl 756 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  (numer `  a
)  =  (numer `  b ) )
65 simprr 758 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  (denom `  a
)  =  (denom `  b ) )
6664, 65oveq12d 6252 . . . . . . . . 9  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  ( (numer `  a )  /  (denom `  a ) )  =  ( (numer `  b
)  /  (denom `  b ) ) )
67 simpll 752 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  a  e.  QQ )
6867, 17syl 17 . . . . . . . . 9  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  a  =  ( (numer `  a )  /  (denom `  a )
) )
69 simplr 754 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  b  e.  QQ )
70 qeqnumdivden 14380 . . . . . . . . . 10  |-  ( b  e.  QQ  ->  b  =  ( (numer `  b )  /  (denom `  b ) ) )
7169, 70syl 17 . . . . . . . . 9  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  b  =  ( (numer `  b )  /  (denom `  b )
) )
7266, 68, 713eqtr4d 2453 . . . . . . . 8  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  a  =  b )
7372ex 432 . . . . . . 7  |-  ( ( a  e.  QQ  /\  b  e.  QQ )  ->  ( ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) )  -> 
a  =  b ) )
7463, 73syl5bi 217 . . . . . 6  |-  ( ( a  e.  QQ  /\  b  e.  QQ )  ->  ( <. (numer `  a
) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  ->  a  =  b ) )
75 fveq2 5805 . . . . . . 7  |-  ( a  =  b  ->  (numer `  a )  =  (numer `  b ) )
76 fveq2 5805 . . . . . . 7  |-  ( a  =  b  ->  (denom `  a )  =  (denom `  b ) )
7775, 76opeq12d 4166 . . . . . 6  |-  ( a  =  b  ->  <. (numer `  a ) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >. )
7874, 77impbid1 203 . . . . 5  |-  ( ( a  e.  QQ  /\  b  e.  QQ )  ->  ( <. (numer `  a
) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  <->  a  =  b ) )
7960, 62, 78syl2anc 659 . . . 4  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
( <. (numer `  a
) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  <->  a  =  b ) )
8079ex 432 . . 3  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( ( a  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  /\  b  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) } )  ->  ( <. (numer `  a ) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  <->  a  =  b ) ) )
8157, 80dom2d 7514 . 2  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  e.  _V  ->  { x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  ~<_  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } ) )
824, 81mpi 18 1  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  ~<_  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   {crab 2757   _Vcvv 3058   <.cop 3977   class class class wbr 4394   {copab 4451    X. cxp 4940   ` cfv 5525  (class class class)co 6234    ~<_ cdom 7472   0cc0 9442   1c1 9443    + caddc 9445    x. cmul 9447    < clt 9578    - cmin 9761   -ucneg 9762    / cdiv 10167   NNcn 10496   2c2 10546   QQcq 11145   ^cexp 12120   sqrcsqrt 13122   abscabs 13123  numercnumer 14367  denomcdenom 14368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-sup 7855  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-n0 10757  df-z 10826  df-uz 11046  df-q 11146  df-rp 11184  df-fl 11879  df-mod 11948  df-seq 12062  df-exp 12121  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-dvds 14088  df-gcd 14246  df-numer 14369  df-denom 14370
This theorem is referenced by:  pellexlem4  35110
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